Restricted Hyperplane Sections of the Cross-Polytope and the Simplex

We give a new proof of Webb’s theorem on maximal central hyperplane sections of the regular $n$-simplex ${\mathrm{\Delta }}_n$, viewed in its standard embedding in ${\mathbb{R}}^{n+1}$. A similar method also yields sharp maximal estimates for non-central sections of ${\mathrm{\Delta }}^n$ whose distance $d$ from the barycenter is small, namely $d<\sqrt{\frac{1}{(n+1)(2n+1)}}\mathrm{.}$ Moreover, we obtain sharp volume estimates for central hyperplane sections of the cross-polytope $B_1^n$ that pass through the barycenter of a facet.